Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-6x-3y &= 3 \\ 7x+4y &= -6\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $4$ and the bottom equation by $3$ $\begin{align*}-24x-12y &= 12\\ 21x+12y &= -18\end{align*}$ Add the top and bottom equations. $-3x = -6$ Divide both sides by $-3$ and reduce as necessary. $x = 2$ Substitute $2$ for $x$ in the top equation. $-6( 2)-3y = 3$ $-12-3y = 3$ $-3y = 15$ $y = -5$ The solution is $\enspace x = 2, \enspace y = -5$.